Definition:Combinatorial Matrix

Definition

A combinatorial matrix is a square matrix specified as:

$a_{i j} = y + \delta_{i j} x$

where $\delta_{i j}$ is the Kronecker delta.

In full, it can be written:

$\begin {bmatrix} x + y & y & \cdots & y \\ y & x + y & \cdots & y \\ \vdots & \vdots & \ddots & \vdots \\ y & y & \cdots & x + y \end {bmatrix}$

It can also be expressed as:

$x \mathbf I_n + y \mathbf J_n$

where:

$\mathbf I_n$ is the unit matrix of order $n$

$\mathbf J_n$ is the square ones matrix of order $n$.

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: Exercises -- Second Set